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Checking Arbitrage in a Three-Strike Call Portfolio

Article Quant Q&A · Author: Abhay Agarwal
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Summary

The document sets up an arbitrage check for calls on the same underlying with strikes 100, 120, and 130 and prices 8, 5, and 3. It represents a portfolio with signed quantities, requires a negative initial cost, and writes its maturity payoff across regions determined by the strikes. Evaluating the payoff at the highest strike gives one constraint on the first two quantities.

The question is how to obtain the further condition that the sum of all three quantities is nonnegative. For underlying prices above the highest strike, the payoff is linear, with that sum as its slope. A negative slope would eventually make the payoff negative as the underlying price rises without bound, violating the nonnegative-payoff requirement. The post gives the setup and question, but does not include a complete portfolio or establish an arbitrage; the slope argument supplies the missing reasoning.

Key ideas

  • An arbitrage portfolio must have negative initial cost and nonnegative payoff in every terminal state.
  • The call portfolio payoff is piecewise linear across the strike intervals.
  • Above the highest strike, the payoff slope equals the sum of the three option quantities.
  • A negative slope would imply negative payoff at sufficiently high underlying prices.

Tags

OptionsArbitrageDerivatives pricing
Full text
# Clarification regarding solution to Call-Options Arbitrage question


# Clarification regarding solution to Call-Options Arbitrage question












I'm having a hard time understanding the solution to the following problem from Dan Stefanica's book "A Primer for the Mathematics of Financial Engineering":

Call options with strikes 100, 120, and 130 on the same underlying asset and with the same maturity are trading for 8, 5, and 3, respectively (there is no bid-ask spread). Is there an arbitrage opportunity present? If yes, how can you make a riskless profit?

To model the situation, we assume $x_1, x_2, x_3$ units of the three options are purchased (where the $x_i$ may be negative as well). Then, the cost of the portfolio is $8x_1 + 5x_2 + 3x_3$, which we want to be negative for an arbitrage opportunity to exist. So we get our first equation

$$8x_1 + 5x_2 + 3x_3 < 0$$

Then we model the payoff of the portfolio at maturity as $$C(T) = x_1 max(S(T) - K_1, 0) + x_2 max(S(T) - K_2, 0) + x_3 max(S(T) - K_3, 0)$$. This gives us a piecewise description of the payoff -

\begin{equation} C(T) = \left\{\begin{array}{lr} 0, & S(T) \le K_1 \\ x_1(S(T) - K_1), & K_1 \le S(T) \le K_2\\ (x_1 + x_2)S(T) - x_1K_1 - x_2K_2, & K_2 \le S(T) \le K_3\\ (x_1 + x_2 + x_3)S(T) - x_1K_1 - x_2K_2 - x_3K_3, & K_3 \le S(T) \end{array}\right. \end{equation}

Each of these should be non-negative if an arbitrage opportunity is to exist. If we set $S(T) = K_3$, then we get $(x_1 + x_2)K_3 - x_1K_1 - x_2K_2 \geq 0$. Substituting the actual numbers in, we get $30x_1 + 10x_2 \geq 0$.

Now, here comes my confusion - the solution states that from the payoff description, we can also come up with a third condition $x_1 + x_2 + x_3 \geq 0$. I'm not sure how to arrive at this condition based on the current information we have. Any thoughts?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.